Download A Crash Course in Topological Groups

A Crash Course in Topological Groups
Iian B. Smythe
Department of Mathematics
Cornell University
Olivetti Club
November 8, 2011
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Topological Groups
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Outline
1
Examples and definitions
2
Basic topological properties
3
Basic “algebraic” properties
4
Further Results
Harmonic analysis on locally compact groups
Descriptive set theory on Polish groups
Hilbert’s Fifth Problem
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Examples and definitions
Motivating examples
A Lie group G is a group, which is also a smooth manifold, such
that the group operations (multiplication and inversion) are
smooth. In particular, G is a topological space such that the group
operations are continuous.
A Banach space X is a complete normed vector space. In
particular, X is an abelian group and a topological space such that
the group operations (addition and subtraction) are continuous.
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Examples and definitions
Main Definition
Definition
A topological group is a group G, which is also a topological space,
such that the group operations,
× : G × G → G, where G × G has the product topology, and
−1
: G → G,
are continuous.
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Examples and definitions
Examples
Lie groups: GLn (R), SLn (R), O(n), U(n), PSL2 (C) ...
The underlying additive group of a Banach space (or more
generally, topological vector space): Rn , Lp (X , µ), C0 , Cc (X ), ...
Groups of homeomorphisms of a ‘nice’ topological space, or
diffeomorphisms of a smooth manifold, can be made into
topological groups.
Any group taken with the discrete topology.
Any (arbitrary) direct product of these with the product topology.
Note that, for example, (Z/2Z)ω is not discrete with the product
topology (it is homeomorphic to the Cantor set).
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Basic topological properties
Homogeneity
A topological group acts on itself by certain canonical
self-homeomorphisms: inversion, left (or right) translation by a
fixed element, and conjugation by a fixed element.
Translation by elements gives a topological group a homogeneous
structure, i.e. we can move from a point h in the group to any
other point k by the homeomorphism g 7→ kh−1 g.
This allows us to infer certain topological information about the
whole group from information at any particular point (such as the
identity).
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Basic topological properties
Homogeneity (cont’d)
Proposition
Let G and H be topological groups.
Denote by NG the set of all open neighbourhoods of e in G. Then,
{gU : g ∈ G, U ∈ NG } is exactly the topology on G.
A group homomorphism ϕ : G → H is continuous on G if and only
if ϕ is continuous at one point of G.
Proof.
If V is any nonempty open set in G, it contains some g ∈ G.
g −1 V ∈ NG and V = g(g −1 V ). This proves the first claim.
We may assume that ϕ is continuous at eG . Let g ∈ G, and W ⊆ H an
open set containing ϕ(g). By continuity at eG , there is U ∈ NG with
ϕ(U) ⊆ ϕ(g)−1 W . So, ϕ(gU) = ϕ(g)ϕ(U) ⊆ W .
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Topological Groups
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Basic topological properties
Homogeneity (cont’d)
Similarly, we can extend many local properties from any particular
point in a topological group to the whole group.
Proposition
Let G be a topological group.
G is locally compact if and only if there is one point of G with a
local basis of compact sets.
G is locally (path) connected if and only if there is one point of G
with a local basis of open, (path) connected sets.
G is locally euclidean if and only if there is one point of G with a
neighbourhood homeomorphic to an open subset of Rn .
G is discrete if and only if there is one point of G which is isolated.
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Topological Groups
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Basic topological properties
Separation Properties
Recall the following terminology:
Definition
Let X be a topological space.
X is T0 if for any pair of distinct points x, y ∈ X , there exists an
open set containing one but not the other.
X is T1 if for any pair of distinct points x, y ∈ X , there are open
sets containing each point, and not containing the other.
X is T2 (Hausdorff) if for any pair of distinct points x, y ∈ X , there
are disjoint open sets containing each.
X is T3 if X is T1 and regular: i.e. if U ⊆ X is an open set
containing x, then there is an open set V with x ∈ V ⊆ V ⊆ U.
In general, T3 =⇒ T2 =⇒ T1 =⇒ T0 , and these implications are strict.
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Topological Groups
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Basic topological properties
Separation Properties (cont’d)
The distinctions between the aforementioned separation properties
dissolve in a topological group.
Lemma
Let G be a topological group. For every U ∈ NG , we have U ⊆ U −1 U.
Proof.
Let g ∈ U. Since Ug is an open neighbourhood of g, Ug ∩ U 6= ∅. That
is, there are u1 , u2 ∈ U with u1 g = u2 , so g = u1−1 u2 ∈ U −1 U.
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Topological Groups
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Basic topological properties
Separation Properties (cont’d)
Proposition
Every topological group G is regular, and if G is T0 , then G is T3 .
Proof.
Let U ∈ NG . By continuity of multiplication and inversion, one can
show that there is V ∈ NG such that VV ⊆ U and V −1 = V . By the
lemma, V ⊆ V −1 V = VV ⊆ U. Hence, G is regular at e, which suffices
by homogeneity.
Suppose G is T0 , and let g, h ∈ G be distinct. We may assume that
there is U ∈ NG such that h ∈
/ Ug. So, hg −1 ∈
/ U, and thus, g ∈
/ U −1 h.
That is, G is T1 and regular, hence T3 .
Iian B. Smythe (Cornell)
Topological Groups
Nov. 8, 2011
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Basic topological properties
Separation Properties (cont’d)
In fact, more is true (with significantly more work):
Theorem
Every topological group G is completely regular, i.e. for any closed
F ⊆ G with g ∈
/ F , there is a continuous function f : G → [0, 1] such
that f (g) = 0 and f (F ) = {1}. If G is T0 , then G is T3 1 (or Tychonoff).
2
Further, a result of Birkhoff and Kakutani shows that topological
groups satisfy the “ultimate” separation property, metrizability, under
very weak assumptions:
Theorem
A topological group G is metrizable if and only if G is T0 and first
countable (i.e. every point has a countable local basis).
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Topological Groups
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Basic “algebraic” properties
“Algebra” with topological groups
The class of all topological groups forms a category where the
morphisms are continuous group homomorphisms.
Isomorphisms in this setting are continuous group isomorphisms
with continuous inverses, and embeddings are continuous group
embeddings which are open onto their images.
Some of the classical constructions from group theory can be
carried over to topological groups in a natural way.
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Topological Groups
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Basic “algebraic” properties
“Algebra” with topological groups (cont’d)
Definition
Let G be a topological group, and N a normal subgroup. The quotient
topological group of G by N is the group G/N together with the
topology formed by declaring U ⊆ G/N open if and only if π −1 (U) is
open in G, where π : G → G/N is the canonical projection.
π : G → G/N is a quotient map in the topological sense, i.e. it is
continuous, open and surjective. However, care must be taken when
forming quotients in light of the following (easy) fact:
Proposition
G/N is Hausdorff if and only if N is a closed normal subgroup of G.
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Topological Groups
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Basic “algebraic” properties
“Algebra” with topological groups (cont’d)
The familiar UMP for quotients holds in this setting:
Proposition
Let G and H be topological groups, ϕ : G → H a continuous group
homomorphism, and N a normal subgroup of G. If N ⊆ ker(ϕ), then
there exists a unique continuous group homomorphism ϕ̃ : G/N → H
such that ϕ = ϕ̃π. That is,
G
π
ϕ
/H
=
ϕ̃
G/N
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Topological Groups
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Basic “algebraic” properties
“Algebra” with topological groups (cont’d)
What about the isomorphism theorems?
Proposition
The First Isomorphism Theorem fails for topological groups.
Proof.
Consider the additive group R with its usual topology, and denote by
Rd the additive group of reals with the discrete topology. The identity
map id : Rd → R is a continuous group homomorphism with trivial
kernel, but Rd / ker(id) = Rd R as topological groups.
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Topological Groups
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Basic “algebraic” properties
“Algebra” with topological groups (cont’d)
Proposition
The Second Isomorphism Theorem fails for topological groups.
Proof.
Consider the additive group R with its usual topology, and the closed
(normal) subgroups N = Z and H = {αn : n ∈ Z}, where α is irrational.
Clearly H ∩ N = {0}, and so H ∼
= H/(H ∩ N) as topological groups; in
particular, both are discrete. However, H + N is a dense subset of R,
so (H + N)/N is a dense subset of the compact group R/Z ∼
= U(1).
In particular, (H + N)/N is not discrete, and (H + N)/N H/(H ∩ N)
as topological groups.
Strangely enough, the Third Isomorphism Theorem holds (exercise).
Iian B. Smythe (Cornell)
Topological Groups
Nov. 8, 2011
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Basic “algebraic” properties
The Connected Component of the Identity
An important, and elementary, example of the interplay between
algebra and topology within a topological group is the connected
component of the identity. Recall from topology:
Definition
Let X be a topological space, and x ∈ X . The connected component of
x in X is the largest connected subset of X containing x.
Proposition
Let G be a topological group, and denote by G0 the connected
component of e in G. Then, G0 is a closed, normal subgroup of G.
Iian B. Smythe (Cornell)
Topological Groups
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Basic “algebraic” properties
The Connected Component of the Identity (cont’d)
Proof.
That G0 is closed is topology (connected components are closed).
We know that e ∈ G0 by definition. Let g, h ∈ G0 . Multiplication on the
right is a homeomorphism, so G0 h−1 is connected, and it contains e
since h ∈ G0 . Thus, G0 h−1 ⊆ G0 , and in particular gh−1 ∈ G0 , showing
that G0 is a subgroup.
Conjugation by any fixed element of G is a homeomorphism which
sends e 7→ e, so G0 is normal in G (in fact, G0 is fully invariant under
continuous endomorphisms of G).
Corollary
Let G be a topological group, then G/G0 is the (totally disconnected)
group of connected components of G.
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Topological Groups
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Further Results
Further results
We will now survey some deeper results on certain classes of
topological groups relevant to analysts, set theorists, and
topologists, respectively.
For the remainder of this talk, all topological groups are assumed
to be T0 , and in particular Hausdorff.
Iian B. Smythe (Cornell)
Topological Groups
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Further Results
Harmonic analysis on locally compact groups
Harmonic analysis on locally compact groups
Definition
A topological group G is locally compact if the underlying space is
locally compact. Equivalently, e has a compact neighbourhood.
The following theorem, due to Haar (1933) and Weil (1940), allows us
to do analysis in a meaningful way on such groups:
Theorem
Every locally compact group G admits a (left) translation invariant
Borel measure, called the (left) Haar measure µ on G. That is, µ is a
measure on the Borel subsets of G such that
µ(gB) = µ(B), for all g ∈ G, and Borel sets B ⊆ G.
Moreover, µ(K ) is finite for every compact K ⊆ G, and µ is unique up
to multiplication by a positive constant.
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Topological Groups
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Further Results
Harmonic analysis on locally compact groups
Harmonic analysis on locally compact groups (cont’d)
The proof establishing the existence of the Haar measure is
nonconstructive, but many examples can be given explicitly.
Lebesgue measure on Rn .
| det(X )|−n dX on GLn (R), where dX is Lebesgue measure on
Rn×n .
Counting measure on any discrete group.
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Further Results
Descriptive set theory on Polish groups
Descriptive Set Theory and Polish Groups
The following important class of spaces is a core object of study in
descriptive set theory :
Definition
A topological space X is said to be Polish if it is separable and
completely metrizable.
Naturally,
Definition
A topological group G is said to be Polish if its underlying space is
Polish.
Examples include Lie groups, separable Banach spaces, countable
discrete groups, and any countable products of these.
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Further Results
Descriptive set theory on Polish groups
Descriptive Set Theory and Polish Groups (cont’d)
Definition
Let X be a topological space.
A ⊆ X is nowhere dense, if the interior of the closure of A is empty.
M ⊆ X is meager if its contained in a countable union of nowhere
dense sets
B ⊆ X has the Baire property if there is an open set U ⊆ X such
that B∆U is meager.
Definition
Let X and Y be topological spaces. A function f : X → Y is Baire
measurable if inverse images of open sets have the Baire property.
It can be shown that every Borel subset of a Polish space has the
Baire property.
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Topological Groups
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Further Results
Descriptive set theory on Polish groups
Descriptive Set Theory and Polish Groups (cont’d)
The following theorem of Pettis (1950) shows that, in some sense,
every homomorphism that we can describe between Polish groups is
continuous.
Theorem
Let G and H be Polish groups. If ϕ : G → H is a Baire measurable
homomorphism, then ϕ is continuous.
Shelah (1984) showed that it is consistent with ZF that every subset of
every Polish space has the Baire property. Hence, it is consistent with
ZF that every homomorphism between Polish groups is continuous,
and constructing non-continuous homomorphisms between such
groups requires use of the Axiom of Choice.
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Topological Groups
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Further Results
Hilbert’s Fifth Problem
Hilbert’s Fifth Problem
In 1900, David Hilbert published a list of what he considered to be the
23 most important problems in mathematics.
The fifth problem on this list was stated as follows: How far Lie’s
concept of continuous groups of transformations is approachable in
our investigations without the assumption of the differentiability of the
functions.
This problem is notably vague and has been understood in many
different ways. The interpretation which received the most attention
was as follows: Is every locally euclidean topological group isomorphic
to a Lie group?
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Further Results
Hilbert’s Fifth Problem
Hilbert’s Fifth Problem (cont’d)
von Neumann (1933) established the positive answer to the previous
question in the case of compact groups, and work of Gleason (1952)
and Montgomery-Zippin (1952) established the general answer:
Theorem
A topological group G is locally euclidean if and only if it is isomorphic
to a Lie group.
Moreover, a locally euclidean group can be (uniquely) endowed with
the structure of a real-analytic manifold.
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Topological Groups
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Further Results
Hilbert’s Fifth Problem
Hilbert’s Fifth Problem (cont’d)
Due to the vagueness of Hilbert’s original question, the story of this
problem is far from over. The following result, due to Goldbring (2010),
concerns spaces having group operations locally around an identity:
Theorem
A local group G is locally euclidean if and only if a restriction of G to an
open neighbourhood of the identity is isomorphic to a local Lie group.
Alternatively, one can also interpret Hilbert’s problem in the form of the
Hilbert-Smith Conjecture:
Conjecture
If G is a locally compact group which acts continuously and effectively
on a connected manifold M, then G is isomorphic to a Lie group.
This question remains open to this day.
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